A. Category theory 89
A.2. Adjunctions and monads
Adjoint functors LetL: C → DandR:D → C be functors. The functor L is called left-adjoint to R resp. the functor R is called right-adjoint to L, if there are transformationsη: IdC →RLandε:LR→IdD, such that the compositions
L(C)−−−−→L(ηC) LRL(C)−−−→εLC L(C) and R(D)−−−−→ηR(D) RLR(C)−−−−−→R(εD) R(D)
are identity morphisms. We write L a R and say that (L, R, η, ε) is an adjunction, the transformationη is calledunit of the adjunction andεis calledcounit of the adjunction.
IfR andR0 are right-adjoint to a functorL, then there is a unique natu-ral isomorphismϕ:R→R0 compatible with the units and counits of the corresponding adjunctions.
We say that (L, R, η, ε) is anadjoint equivalence, ifη andεare natural isomorphisms. The functorRis then a quasi-inverse ofL; every equivalence of categories is part of an adjoint equivalence.
Monads LetT:C → Cbe a functor and letµ: T◦T →Tandη: IdC →T be natural transformations. The triple (T, µ, η) is called amonad on C, if the equalities
µA◦T(µA) =µA◦µT A and µA◦ηT A= idT A=µA◦T(ηA)
hold for every object A in C. We also say that T is a monad without mentioning the transformations µ and η, called multiplication and unit respectively.
Amodule overT orT-module is a pair (A, r) consisting of an objectA inC and a morphismr:T A→A, such that
r◦µA=r◦T(r) and r◦ηA= idA.
If (A, rA) and (B, rB) are T-modules a morphism f:A → B is called T-linear, if
f◦rA=rB◦T(f).
TheEilenberg-Moore category CT of a monadT has as objectsT-modules and as morphisms T-linear morphisms; the composition of morphisms is inherited fromC.
Monadic functors Theforgetful functor UT: CT → C sending (A, rA) to Ais part of an adjunction (UT,FT, ηT, εT).
The right-adjoint functor of UT is the free functor FT: C → CT, which sends an object A to the T-module (T A, µA) and a morphism f to the T-linear mapT(f). The unit of T defines the unit ηT of the adjunction and the counitηT has the componentsεT(A,r
A)=rA.
The monad of an adjunction (L, R, η, ε) is given by the functorT :=
RL with multiplication µA := R(εLA) :RLRL(A) → RL(A) and unit ηA:A→RL(A).
Let G: D → C be a functor, that is part of an adjunction (F, G, η, ε) and byT its associated monad onC. Thecomparison functor K:D → CT is given byK(D) := (G(D), G(εD)).
A functorG:D → C is called monodic, if it has a left-adjoint F:C → D and the comparison functorK:D → CT is an equivalence of categories.
A.3. Monoidal categories
Definition A.3.1 A monoidal category is a 6-tuple (C,⊗,1,a,l,r) con-sisting of a categoryC, a functor⊗: C × C → C(tensor product), an object
1inC, a natural isomorphisma: ⊗ ◦(⊗ ×IdC)→ ⊗ ◦(IdC× ⊗) (associa-tor), a natural isomorphisml:1⊗ →IdC (left unit isomorphism) and a natural isomorphism r: ⊗1→IdC (right unit isomorphism), such that the following diagrams commute for all objectsU, V, W andX inC:
((U⊗V)⊗W)⊗X
(U⊗V)⊗(W ⊗X)
U⊗(V ⊗(W ⊗X)) (U⊗(V ⊗W))⊗X U⊗((V ⊗W)⊗X)
aU⊗V,W,X 44 aU,V,W⊗X
**
aU,V,W⊗idX aU,V⊗W,X
// idU⊗aV,W,XGG
(U⊗1)⊗V U⊗(1⊗V)
(U⊗1)⊗V
aU,1,V //
rU⊗idV )) uu idU⊗lV
The category is calledstrict, ifa,landr are the identity transformations.
Definition A.3.2 Let C and D be monoidal categories, let F:C → D be a functor, let F0: F1C → 1D and F0: 1D → F be morphisms and F2:F◦ ⊗ →F⊗F andF2:F⊗F→F◦ ⊗transformations.
1. The triple (F, F2, F0) is calledlax monoidal functor and F2 andF0
are called themonoidal structure ofF, if for all objectsU, V, W inC the diagrams
(F U⊗F V)⊗F W F U⊗(F V ⊗F W) F(U⊗V)⊗F W F U⊗F(V ⊗W) F((U⊗V)⊗W) F(U⊗(V ⊗W))
aF U,F V,F W //
F(aU,V,W)
//
F2(U,V)⊗idF W
F2(U⊗V,W)
idF U⊗F2(V,W)
F2(U,V⊗W)
F(1⊗U) F U F(U⊗1)
F1⊗F U 1⊗F U F U⊗1 F U⊗F1
F(lU) // oo F(rU)
F2(1,U)OO
F0⊗id
oo lF U ??
rF U
__
id⊗F0//
F2(U,1)
OO
commute. The monoidal functor is calledstrong monoidal or simply monoidal, ifF2andF0are isomorphisms. It is calledstrict, ifF2and F0 are identities.
2. The triple (F, F2, F0) is calledoplax monoidal functor andF2 and F0are called the(op)monoidal structure ofF, if the diagrams above
commute after changingF2toF2andF0toF0(and of course chang-ing the direction of those arrows). If is called strong (op)monoidal resp.strict, ifF2 andF0are isomorphisms resp. identities.
3. LetF andG be lax monoidal resp. oplax monoidal functors. If de-fined, the composition F ◦G is an (op)lax monoidal functor with monoidal structure
(F G)2(U, V) :=F(G2(U, V))◦F2(GU, GV) and (F G)0:=F(G0)◦F0
resp. opmonoidal structure
(F G)2(U, V) :=F2(GU, GV)◦F(G2(U, V)) and (F G)0:=F0◦F(G0).
Definition A.3.3 Let F and G be (op)lax monoidal functors and let α:F →Gbe a natural transformation. If the equalities
αU⊗V ◦F2(U, V) =G2(U, V)◦(αU⊗αV) and α1F0=G0 (A.1) hold for all objectsU andV inCwe say thatαis amonoidal transforma-tion. We callαanopmonoidal transformation, if the equalities
(αU ⊗αV)◦F2(U, V) =G2(U, V)◦αU⊗V (A.2) hold for all objectsU andV inC. By abuse of language we call opmonoidal transformations sometimes also monoidal.
Lemma A.3.4 Let (F, G, η, ε) be an adjunction. The following statements hold, as well as the dual statements, i.e. exchangeF andGand the words lax and oplax.
1. IfG is a lax monoidal functor there is an oplax monoidal structure onF.
2. If F and G are lax monoidal functors and η and ε are monoidal transformations, thenF is a strong monoidal functor.
3. IfF is strong monoidal, there is a unique lax monoidal structure on G, such thatη andεare monoidal transformations.
Sketch of proof.
1. The morphism set Hom(X⊗Y, G(F X⊗F Y)) contains the morphism f :=G2(F X, F Y)◦(ηX⊗ηX). Define F2(X, Y) as the image of f under the isomorphism
Hom(X⊗Y, G(F X⊗F Y))∼= Hom(F(X⊗Y), F X⊗F Y), i.e.F2(X, Y) =εF X⊗F Y ◦F(f). DefineF0:=ε1◦F(G0).
2. Under the additional assumptions onF, η andεone proves thatF2 is inverse toF2 andF0 is inverse toF0.
3. Assume that G2(X, Y) defines a monoidal structure on G. Since η and ε are monoidal transformations we conclude from 2. that G2(X, Y) defines an oplax monoidal structureF2(X, Y), whose com-ponents are inverse toF2(X, Y). HenceF2(X, Y) =F2−1(X, Y) and G2(X, Y) is the monoidal structure obtained from the dual statement of 1.
Corollary A.3.5 Let (F, G, η, ε) be an adjoint equivalence. The functor F is strong monoidal, iffGis strong monoidal.
Remark A.3.6 Usually one can assume, without loss of generality, that one works in a strict monoidal category due to the fact that every monoidal categoryC is equivalent, as a monoidal category, to a strict monoidal cat-egoryCstr. See for example SectionXI.5 in [Kas95].
Definition A.3.7 LetCbe a strict monoidal category and letXbe an ob-ject inC. Aleft dual object forXis an object∨X together with morphisms evX:∨X⊗X→1(evaluation) and coevX:1→X⊗∨X (coevaluation), such that
(evX⊗id∨X)(id∨X⊗coevX) = id∨X and (idX⊗evX)(coevX⊗idX) = idX.
Analogously one defines aright dual object as an objectX∨ together with morphismseveX:X⊗X∨→1and coeve X:1→X∨⊗X fulfilling
(idX∨⊗eveX)(coeve X⊗idX∨) = idX∨ and (eveX⊗idX)(idX⊗coev) = ide X.
If every objectX inC has a left resp. right dual, the category is calledleft rigid resp.right rigid. The category C is calledrigid, if it is left and right rigid.
IfX0 andX00are both left or right dual to X, there is a unique isomor-phism X0 → X00 compatible with evaluation. Let C be (say) right rigid.
If we choose exactly one dual object for each objectX, we get a functor ( )∨:Cop→ C by defining forf:X →Y the morphism
f∨:= (id⊗eveY)(id⊗f⊗id)(coeve X⊗id).
Similarly, one defines a functor ∨( ) : Cop → C. The functors ∨( ) and ( )∨ are strong monoidal Cop → C⊗op where C⊗op denotes the monoidal category with opposed tensor productX⊗opY :=Y ⊗X.
Definition A.3.8 Let C be a rigid category and α: ∨( ) →( )∨ a nat-ural isomorphism. It is called pivotal structure on C, if it is a monoidal isomorphism. For an endomorphism f: X → X in a pivotal category C one defines theleft trace tr`(f) and theright tracetrr(f) as
tr`(f) := evX◦(α−1X ⊗f)◦coeve X
tr`(f) :=eveX◦(f⊗αX)◦coevX.
A pivotal structure is calledspherical, if left and right trace coincide for all endomorphisms ofC.
Definition A.3.9 LetCbe a monoidal category andcX,Y:X⊗Y →Y⊗X a family of morphisms, natural inX, Y ∈ Ob(C). The familycX,Y is called prebraiding, if the following two hexagons commute for allX, Y, Z∈ Ob(C)
X⊗(Y ⊗Z)
(X⊗Y)⊗Z Z⊗(X⊗Y) (Z⊗X)⊗Y X⊗(Z⊗Y) (X⊗Z)⊗Y
a−1 ?? cX⊗Y,Z //
a−1
id⊗cY,Z a−1 // cX,Z?? ⊗id
(X⊗Y)⊗Z
X⊗(Y ⊗Z) (Y ⊗Z)⊗X Y ⊗(Z⊗X)
(Y ⊗X)⊗Z Y ⊗(X⊗Z)
a ?? cX,Y⊗Z //
a
cX,Y⊗id a // id⊗c?? X,Z
Ifc is a natural isomorphism the categoryC is called a braided category withbraiding c.
LetCandDbe braided categories. A monoidal functorF:C → Dis called braided, if
F(cX,Y)◦F2(X, Y) =F2(Y, X)◦cF X,F Y .
Definition A.3.10 LetCbe a braided category. Thesymmetric centerof Cor the subcategory oftransparent objects ZsyminCis defined as the full subcategory ofCcontaining the objectsX, such thatcY,X◦cX,Y = idX⊗Y for allY ∈ Ob(C).
A.4. Modular categories
Letk be a field. We call a category C k-linear, ifC has a zero object, all finite sums and every Hom set is ak-vector space such that the composition
◦of morphisms isk-linear in each variable.
LetIbe an indexing set and{Ci}i∈I a family ofk-linear categories. The direct sumL
i∈ICiis defined as the following category: objects are families (Xi)i∈I whereXi is an object inCi and only finitely manyXiare not the zero object, the morphism set Hom((Xi)i∈I,(Yi)i∈I) is given by the vector spaceL
i∈IHom(Xi, Yi).
An object X in a k-linear category is called simple, if the Hom-space C(X, X) is one dimensional.
Ak-linear abelian category is calledfinite, if
• there are, up to isomorphism, only finitely many simple objects,
• every Hom spaceC(X, Y) is finite dimensional,
• every object has finite length,
• the categoryC has enough projectives.
Atensor category is a rigid, monoidal category C which isk-linear, the unit object1is simple and the functor ⊗isk-linear in each variable.
A finite tensor category C is called fusion category, if every object is isomorphic to a finite sum of simple objects.
A premodular category is a braided fusion category together with a spherical structure.
A premodular categoryC is called modular, if Zsym(C) is equivalent to the categoryvectkof finite dimensional vector spaces.
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Hiermit erkl¨are ich an Eides statt, dass ich die vorliegende Dissertations-schrift selbst verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.
Hamburg, den 08.05.2014 Unterschrift
Monoidale Kategorien treten auf als Darstellungskategorien von Hopf-Algebren und anderen algebraischen Strukturen. Sie spielen unter anderem eine zentrale Rolle in Darstellungstheorie und topologischer Feldtheorie. In dieser Dissertation werden zwei verschiedene Konstruktionen durchgef¨uhrt, die aus diesen Zusammenh¨angen heraus motiviert sind. Im Besonderen spielen dabei algebraische und darstellungstheoretische Strukturen inner-halb dieser monoidalen Kategorien eine wichtige Rolle. ¨Uber diese gibt das erste Kapitel einen kurzen ¨Uberblick.
Kapitel 2 besch¨aftigt sich mit Yetter-Drinfel’d Moduln ¨uber Hopf-Alge-bren in verzopften Kategorien und stellt neue Resultate f¨ur die folgenden Kapitel bereit. Ein besonderer Schwerpunkt liegt dabei auf der Beschrei-bung eines Isomorphismus Ω :AAYD(C)→AA∨∨YD(C) von verzopften Kate-gorien, wobei, falls existent,A∨die duale Hopf-Algebra vonAbezeichnet.
Im dritten Kapitel erinnern wir, f¨ur eine diskrete Gruppe Γ, an die Definition einer Γ-verzopften Kategorie und einer quasi-triangul¨aren Hopf Γ-Koalgebra.
Im vierten Kapitel wird die erste Konstruktion dieser Arbeit vorgestellt:
Zu einer beliebigen monoidale KategorieCmit schwacher Gruppenwirkung durch eine Gruppe Γ, konstruieren wir eine Γ-verzopfte KategorieZΓ(C), deren neutrale Komponente das Drinfel’d Zentrum von C ist. Ist C die Kategorie von Moduln ¨uber einer Hopf-Algebra H, so lassen sich die ho-mogenen Komponenten vonZΓ(C) als getwistete Yetter-Drinfel’d Moduln
¨
uberH beschreiben. F¨ur eine endlich-dimensionale Hopf-Algebra H ver-allgemeinert unsere KategorieZΓ(C) eine Hopf-algebraische Konstruktion von Virelizier.
Das letzte Kapitel behandelt die zweite Konstruktion dieser Arbeit. Mo-tiviert durch Arbeiten von Heckenberger und Schneider f¨uhren wir den Be-griff einer partiellen Dualisierung von Hopf-Algebren in verzopften Kate-gorien ein. Eine wichtige Idee in dieser Arbeit ist es, im Rahmen verzopfter Kategorien algebraisch aufw¨andige Rechnungen mit Smash-Produkten kon-zeptionell zu vereinfachen. Ausgehend von einer Hopf-AlgebraH in einer verzopften Kategorie C mit einer Hopf-Unteralgebra A und einer Hopf-Algebraprojektion π:H → A konstruieren wir die partielle Dualisierung H0 vonH bez¨uglichA. Die Hopf-AlgebrenH undH0sind im Allgemeinen weder isomorph noch Morita-¨aquivalent. Wir zeigen aber, dass die Kat-egorienHHYD(C) und HH00YD(C) immer isomorph als verzopfte Kategorien sind. Dar¨uber hinaus ist die partielle Dualisierung vonH0 bzgl. der Unter-algebraA∨ wieder isomorph zuH.